A Worst-Case Approach to Option Pricing in Crash-Threatened Markets

Transcription

1 A Worst-Case Approach to Option Pricing in Crash-Threatened Markets Christoph Belak School of Mathematical Sciences Dublin City University Ireland Department of Mathematics University of Kaiserslautern Germany Joint work with Olaf Menkens and Christian Ewald Dublin City University December 13, / 14

2 Objective Worst-Case Option Pricing in Crash-Threatened Markets 2 / 14

3 Objective Worst-Case Option Pricing in Crash-Threatened Markets The objective of this research project is to: Determine a fair price of a so-called derivative, i.e. a financial instrument, whose value is based on one or more underlying assets. 2 / 14

4 Objective Worst-Case Option Pricing in Crash-Threatened Markets The objective of this research project is to: Determine a fair price of a so-called derivative, i.e. a financial instrument, whose value is based on one or more underlying assets. We extend the standard market model and allow the possibility of crashes in the underlying asset. 2 / 14

5 Objective Worst-Case Option Pricing in Crash-Threatened Markets The objective of this research project is to: Determine a fair price of a so-called derivative, i.e. a financial instrument, whose value is based on one or more underlying assets. We extend the standard market model and allow the possibility of crashes in the underlying asset. We assume that investors are aware of the crash threat and take the worst-case crash scenario into account when pricing the derivative. 2 / 14

6 An Example: European Call Option We assume that our market consists of two underlying assets: A money market account M t A stock S t 3 / 14

7 An Example: European Call Option We assume that our market consists of two underlying assets: A money market account M t A stock S t We want to price a European Call Option The option gives its owner the right (but not the obligation) to buy the stock at some future time T for a price of K. The value of the option at time T is therefore max{s T K, 0}. In particular, the option must have a strictly positive value for all times t < T. 3 / 14

8 An Example: European Call Option We assume that our market consists of two underlying assets: A money market account M t A stock S t We want to price a European Call Option The option gives its owner the right (but not the obligation) to buy the stock at some future time T for a price of K. The value of the option at time T is therefore max{s T K, 0}. In particular, the option must have a strictly positive value for all times t < T. Question: So how do we determine the price of the option at time t? 3 / 14

9 An Example: European Call Option Idea: If at time t, we can setup a portfolio which guarantees a payoff of max{s T K, 0} at time T, then a fair price of the option would be the amount of money needed to setup such a portfolio. 4 / 14

10 An Example: European Call Option Idea: If at time t, we can setup a portfolio which guarantees a payoff of max{s T K, 0} at time T, then a fair price of the option would be the amount of money needed to setup such a portfolio. Let π t denote the amount of total wealth X t invested in the stock. In a simple Black-Scholes model, the investor s wealth is given by dx t = [r(x t π t ) + π t µ]dt + π t σdw t, X 0 = x. Here r denotes the interest rate, µ the drift of the stock and σ the volatility of the stock. 4 / 14

11 An Example: European Call Option Idea: If at time t, we can setup a portfolio which guarantees a payoff of max{s T K, 0} at time T, then a fair price of the option would be the amount of money needed to setup such a portfolio. Let π t denote the amount of total wealth X t invested in the stock. In a simple Black-Scholes model, the investor s wealth is given by dx t = [r(x t π t ) + π t µ]dt + π t σdw t, X 0 = x. Here r denotes the interest rate, µ the drift of the stock and σ the volatility of the stock. Putting the pieces together, we want to find a solution (X t, π t ) of the following Backward Stochastic Differential Equation (BSDE): dx t = [r(x t π t ) + π t µ]dt + π t σdw t, X T = max{s T K, 0}. 4 / 14

12 An Example: European Call Option One can show that the BSDE dx t = [r(x t π t ) + π t µ]dt + π t σdw t, X T = max{s T K, 0}, has an explicit solution. 5 / 14

13 An Example: European Call Option One can show that the BSDE dx t = [r(x t π t ) + π t µ]dt + π t σdw t, X T = max{s T K, 0}, has an explicit solution. More interesting for us: The solution can also be described as the unique solution of the Partial Differential Equation 0 = V t (t, s) + rsv s (t, s) σ2 s 2 V ss (t, s) rv (t, s), V (T, s) = max{s K, 0}. 5 / 14

14 Adding Crashes to the Equation We now assume that at some unknown, possibly random time τ, the stock crashes by a fraction of β: S τ = (1 β)s τ. Important: They crash may or may not occur. However, there is at most one crash. 6 / 14

15 Adding Crashes to the Equation We now assume that at some unknown, possibly random time τ, the stock crashes by a fraction of β: S τ = (1 β)s τ. Important: They crash may or may not occur. However, there is at most one crash. Question: How does this affect the price of the European Option? 6 / 14

16 Adding Crashes to the Equation We now assume that at some unknown, possibly random time τ, the stock crashes by a fraction of β: S τ = (1 β)s τ. Important: They crash may or may not occur. However, there is at most one crash. Question: How does this affect the price of the European Option? In this setting, the payoff max{s T K, 0} can no longer be perfectly replicated. 6 / 14

17 Adding Crashes to the Equation We now assume that at some unknown, possibly random time τ, the stock crashes by a fraction of β: S τ = (1 β)s τ. Important: They crash may or may not occur. However, there is at most one crash. Question: How does this affect the price of the European Option? In this setting, the payoff max{s T K, 0} can no longer be perfectly replicated. The next best idea for the price is to find the minimal initial wealth required, so that our portfolio value X T dominates the payoff max{s T K, 0}. 6 / 14

18 The Worst-Case Price We define the Worst-Case Price of the European Option as 7 / 14

19 The Worst-Case Price We define the Worst-Case Price of the European Option as the minimum initial capital x required 7 / 14

20 The Worst-Case Price We define the Worst-Case Price of the European Option as the minimum initial capital x required to ensure X T max{s T K, 0} 7 / 14

21 The Worst-Case Price We define the Worst-Case Price of the European Option as the minimum initial capital x required to ensure X T max{s T K, 0} no matter which crash scenario (τ, β) occurs. 7 / 14

22 The Worst-Case Price We define the Worst-Case Price of the European Option as the minimum initial capital x required to ensure X T max{s T K, 0} no matter which crash scenario (τ, β) occurs. Let s translate this into BSDE language! We need 7 / 14

23 The Worst-Case Price We define the Worst-Case Price of the European Option as the minimum initial capital x required to ensure X T max{s T K, 0} no matter which crash scenario (τ, β) occurs. Let s translate this into BSDE language! We need dx t = [r(x t π t ) + π t µ]dt + π t σdw t, X T max{s T K, 0}, and 7 / 14

24 The Worst-Case Price We define the Worst-Case Price of the European Option as the minimum initial capital x required to ensure X T max{s T K, 0} no matter which crash scenario (τ, β) occurs. Let s translate this into BSDE language! We need dx t = [r(x t π t ) + π t µ]dt + π t σdw t, X T max{s T K, 0}, and X t βπ t X t, where X t denotes the solution in the crash-free model. 7 / 14

25 The Worst-Case Price We define the Worst-Case Price of the European Option as the minimum initial capital x required to ensure X T max{s T K, 0} no matter which crash scenario (τ, β) occurs. Let s translate this into BSDE language! We need and dx t = [r(x t π t ) + π t µ]dt + π t σdw t, X T max{s T K, 0}, X t βπ t X t, where X t denotes the solution in the crash-free model. Among all solutions, we look for the one which is minimal! 7 / 14

26 Results One can show: If at least one solution of the constrained BSDE dx t = [r(x t π t ) + π t µ]dt + π t σdw t, X T max{s T K, 0}, X t X t + βπ t, exists, then there exists a minimal solution. 8 / 14

27 Results One can show: If at least one solution of the constrained BSDE dx t = [r(x t π t ) + π t µ]dt + π t σdw t, X T max{s T K, 0}, X t X t + βπ t, exists, then there exists a minimal solution. This minimal solution solves the PDE { max V t (t, s) + rsv s (t, s) σ2 s 2 V ss (t, s) rv (t, s), } V (t, (1 β)s) V (t, s) + βsv s (t, s) = 0, { max max{s K, 0} V (T, s), } V (T, (1 β)s) V (T, s) + βsv s (T, s) = 0. 8 / 14

28 Numerical Examples: Payoff Function 9 / 14

29 Numerical Examples: t = 0.5, β = / 14

30 Numerical Examples: t = 1, β = / 14

31 Conclusions There are some issues we need to address: How do we interpret the new payoff function at terminal time? Try to calibrate the model to market data. Extend the results to more complicated options. How to solve PDEs of the form { } max L (2) V, L (1) 1 V, L(1) 2,... = 0. efficiently? 12 / 14

32 Any Questions?? 13 / 14

33 Thank you for your attention!!! 14 / 14